### Two Quick Puzzles

 Follow @ProbabilityPuz The following are two puzzles which look tough at first but have quick and really elegant solutions.

Q1: Ants on a wire:
A large number of ants are on a wire of length $$L$$. All ants start moving randomly, either right or left with a fixed velocity $$V$$. If they collide they turn around and move in the opposite direction. Ants at the ends of the wire fall off. What is the time taken for all ants to fall off the wire?

Q2: The Unruly Passenger:
Several passengers are in a queue to board a plane. The first passenger in the queue is an unruly one and chooses a seat at random. Subsequent passengers take their allotted seat if it is unoccupied or pick a seat at random if it is occupied. What is the probability that the last passenger gets to sit on his allotted seat?
Statistics: A good book to learn statistics A1: This seemingly complex problem has an elegantly simple solution. The fact that they collide and turn around is the same as if they walked through each other! See figure below

Once this is insight sinks in, the average time taken for all ants to fall off the wire can be easily calculated. It is the same as the time an ant takes to move from one end of the wire to the other end. This works out to $$\frac{L}{V}$$.

Effective Java (2nd Edition) A2: You absolutely do not want to consider the various ways a large number of passengers can fill up an equally large number of seats. Bear in mind that the only unruly passenger is the first one, and what we want to know is the probability that the last passenger gets to sit on his seat. The last passenger will face exactly two scenarios, either he gets his seat or not. He will get his seat if the first passenger picks his allotted seat which happens with a probability $$\frac{1}{2}$$

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

1. I am confused. The first passenger picks his seat with probability 1/n (n == num passengers), not 1/2, right?

1. The first passenger picks his seat at random.

2. Suppose there are 100 seats on the airplane. The unruly passenger sits in his assigned seat 1/100 times and chooses someone else's seat 99/100 times. So when we come to the last person, there's a 1/100 chance his seat is unoccupied, and a 99/100 chance that the unruly passenger caused a ripple effect and the last person is stuck in a seat not assigned to him. Right? Generalize for n.

3. I'm not satisfied with A2 either. I offer the following argument: all of the passengers would behave in exactly the same way had the labels on the first and last passengers' seats been switched (or removed altogether). In other words, the given problem is symmetric with respect to these two seats. Hence the probabilities of the events "last passenger sits in first passenger's seat" and "last passenger sits in last passenger's seat" are equal. We know that one of these events must occur, since every seat other than those two must be occupied by the time the last passenger boards. Therefore the two probabilities are both 1/2.

### The Best Books to Learn Probability

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own Discovering Statistics Using R This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing. Fifty Cha

### Fun with Uniform Random Numbers

Q: You have two uniformly random numbers x and y (meaning they can take any value between 0 and 1 with equal probability). What distribution does the sum of these two random numbers follow? What is the probability that their product is less than 0.5. The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists A: Let z = x + y be the random variable whose distribution we want. Clearly z runs from 0 to 2. Let 'f' denote the uniform random distribution between [0,1]. An important point to understand is that f has a fixed value of 1 when x runs from 0 to 1 and its 0 otherwise. So the probability density for z, call it P(z) at any point is the product of f(y) and f(z-y), where y runs from 0 to 1. However in that range f(y) is equal to 1. So the above equation becomes From here on, it gets a bit tricky. Notice that the integral is a function of z. Let us take a look at how else we can simply the above integral. It is easy to see that f(z-y) = 1 when (

### The Best Books for Linear Algebra

The following are some good books to own in the area of Linear Algebra. Linear Algebra (2nd Edition) This is the gold standard for linear algebra at an undergraduate level. This book has been around for quite sometime a great book to own. Linear Algebra: A Modern Introduction Good book if you want to learn more on the subject of linear algebra however typos in the text could be a problem. Linear Algebra (Dover Books on Mathematics) An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book. Linear Algebra Done Right (Undergraduate Texts in Mathematics) A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra. Linear Algebra, 4th Ed